The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B

51with x k∗denoting the midpoint of τ k . Alternatively, we can setg k = 1 3g(x k 1) + g(x k 2) + g(x k 3) .However, since we deal with functions in Lebesgue and Sobolev spaces, where we do notdistinguish between functions differing on a measure zero set, the above mentioned approximationsmay prove impossible. Hence, we introduce the projection P ψ : L 2 (∂Ω) → T ψ (∂Ω)minimizing the error∥g − P ψ g∥ L 2 (∂Ω),i.e.,P ψ g := arg min ∥g − g ψ∥ Lg ψ ∈T ψ (∂Ω) 2 (∂Ω).To find the coefficients g kf : R E → R as E f(g 1 , . . . , g E ) := g − 2g k ψ kk=1= ⟨g, g⟩ L 2 (∂Ω) − 2For the partial derivatives of f we getfor a real-valued function g we introduce the functionL 2 (∂Ω)=g −Eg k ψ k , g −k=1Eg k ⟨g, ψ k ⟩ L 2 (∂Ω) +k=1∂f∂g j= −2⟨g, ψ j ⟩ L 2 (∂Ω) + 2Moreover, for the Hessian of f we haveH f [j, i] =EEg k ψ kk=1Eg kk=1 l=1Eg k ⟨ψ k , ψ j ⟩ L 2 (∂Ω).k=1∂2 f∂g j ∂g i= 2⟨ψ i , ψ j ⟩,L 2 (∂Ω)g l ⟨ψ k , ψ l ⟩ L 2 (∂Ω).which is the double Gram matrix corresponding to the basis of T ψ (∂Ω). Hence, the Hessianis positive definite and to find the minimum of f we set ∂f∂g j= 0 to obtain the system oflinear equationsEg k ⟨ψ k , ψ j ⟩ L 2 (∂Ω) = ⟨g, ψ j ⟩ L 2 (∂Ω) for j ∈ {1, . . . , E}k=1with a unique solution g = [g 1 , . . . , g E ] T ∈ R E . Because∆ k for k = j,⟨ψ k , ψ j ⟩ L 2 (∂Ω) =0 otherwise,